A Mathematician and His Mathematical Work: Selected Papers by Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

Those chosen papers of S.S. Chern talk about subject matters similar to critical geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles

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A simplicia1 subdivision L,, of L, and an almost semilinear realization HrL,,O=L:,O of L, in R" through the subdivision L,,, such that the following conditions are satisfied: 1". L,,,=K,x ( l ) + K , x ( 2 ) + C 6 ; X [l, 21. iC1, 2". H,(T~x ( j ) ) = T j ( T j ) ,r j C K j , j=1, 2 and for ;Elo, R(Zix [ l , 2 ] ) is a simple broken line li. 3". o = H,-I. 4". x [ 1,2] n Rr(Zjx [ 1,2])=@. For the construction let us first draw in R" for each iC], a simple broken line 1; joining x ( i ) ) ,j = 1, 2, such that these l; together with T,K,+T,K, form an almost euclidean complex.

The projection by A: R(b x r) = b * r Then by the definition of Hence we have Theorem 6. *). qr we have evidently be the covering projection of = $=, n* #a R* @" = p, ( m even 6"', Denote ( m odd) > 0) A ' pT = ,qm) qT. g" on K", then (17) (18) in which pz denotes reduction mod 2. Suppose for the moment 6"'= 0 (cf. the remark above), then (17) and (18) may be reduced simply to x*@"'=O, 37 m>O. (19) 266 $ 4 . THEREALIZABILITY OF ANY COCYCLE IN THE IMBEDDING CLASSES W e have proved that the m-dimensional imbedding cochains FT (or pT) of K are all cocycles and belong to one and the same cohomology class 6'"E g"''* ( K ) (or 0"' E H"" ( K , Zcm)).

21 This page intentionally left blank SCIENTIA SINICA Vol. VII, No. 3, 1958 MATHEMATICS ON THE REALIZATION OF COMPLEXES IN EUCLIDEAN SPACES I" ABSTRACT I t was early known that a n y n-dimensional abstract complex may be realized in a ( 2 n + 1)-dimensional euclidean space RZnf'. From this theorem, whose proof is quite simple, i t follows that the ( 2 n + l ) -dimensional euclidean space contains in reality all imaginable n-dimensional complexes. 2+1, is a problem much more difficult which cannot, i t seems, be solved completely i n the near future.

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